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Question Number 203544 by laxmanb2111 last updated on 21/Jan/24

$${\int }_0^{1_{}}\frac{dx}{{(1-x^6)}^{\frac{1}{6}}}=\frac{\pi }{3}$$

Answered by DwaipayanShikari last updated on 23/Jan/24

$${\int }_0^1\frac{1}{{(1-x^n)}^{1/n}}dx$$$$=\frac{1}{n}{\int }_0^1{u}^{\frac{1}{n}-1}{(1-u)}^{-1/n}du$$$$=\frac{1}{n}B(\frac{1}{n},1-\frac{1}{n})$$$$=\frac{1}{n}\mathrm{\Gamma }\left(\frac{1}{n}\right)\mathrm{\Gamma }(1-\frac{1}{n})=\frac{\pi }{nsin\left(\pi /n\right)}$$$$n=6$$$$=\frac{\pi }{3}$$

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